(Thanks to Michael Schmitz for pointing me in the right direction for
answering this question)

You can; however, I strongly recommend against
doing so. There are various reasons: two of the most important are
varying file locations and C library conflicts. I speak from
experience that you don't want to mix files (I'm
still cleaning up old a.out and libc5 files from my system).

Here's a procedure that should work well for doing an installation
from scratch, while keeping your old user files available. This is
not well-tested or anything, but should help you get the idea of what
you need to do:

Print out your /etc/passwd and /etc/group files (you may need them later). You
should also print out your network configuration information.

Back up your Linux partitions; if you are installing on a clean disk
(i.e. not the one you're using now), you can forgo this and the next
two steps.

If you have a home partition with all of
your user directories on it, keep it around. Otherwise, make a tar
file of your home directories tree and copy that somewhere safe (a
non-Linux partition should be fine); make sure you use the "-p" flag
with tar to retain all of the permissions.

Repartition (if you want). But don't clobber home if you already had one.

Add the actual user accounts from your old /etc/passwd
file to your new system, using your distribution's user adding utility
(for Debian, it's useradd). Set the passwords to each
account (or disable them). If you had any special groups (besides a
users or staff group), you may want to add them as well.

Mount your home tree at /home (if it was
on a partition). If you have a tar file of the home tree, figure out where you're going to put
it (/var/home might be a good choice,
although a separate partition would be preferable), untar the home tree there (again using the "-p" flag),
and make a symlink (if necessary) from /home to the root of your new home tree.

If you didn't use the tar file: cd to /home and change the ownership of the users'
files to their new users. For example, for a user named bob (in a
group named users), chmod -R bob.users
/home/bob. If people have interspersed files, you may need
to do a find operation to get the
permissions straight (refer to your old password and group files if
necessary).

This outline should at least point you in the right direction; let me
know if you have any suggestions for improving these instructions.